Chapter 2
Function Spaces and Matrices
Abstract This chapter refreshes such necessary algebraic knowledge as will be
needed in this book. It introduces function spaces, the meaning of a linear operator, and the properties of unitary matrices. The homomorphism between operations
and matrix multiplications is established, and the Dirac notation for function spaces
is defined. For those who might wonder why the linearity of operators need be considered, the final section introduces time reversal, which is anti-linear.
Contents
2.1
Function Spaces ...................................
1 1
2 . 2
L i n e a rO p e r a t o r sa n dT r a n s f o r m a t i o nM a t r i c e s ...................
1 2
2 . 3
U n i t a r yM a t r i c e s ...................................
1 4
2 . 4
T i m eR e v e r s a la sa nA n t i - l i n e a rO p e r a t o r ......................
1 6
2 . 5
P r o b l e m s .......................................
1 9
R e f e r e n c e s...........................................
1 9
2.1 Function Spaces
In the first chapter, we saw that if we wanted to rotate the 2p x function, we automatically also needed its companion 2p y function. If this is extended to out-of-plane
rotations, the 2p z function will also be needed. The set of the three p-orbitals forms
a prime example of what is called a linear vector space. In general, this is a space
that consists of components that can be combined linearly using real or complex
numbers as coefficients. An n-dimensional linear vector space consists of a set of n
vectors that are linearly independent. The components or basis vectors will be denoted as f l , with l ranging from 1 to n. At this point we shall introduce the Dirac notation [1] and rewrite these functions as |f l , which characterizes them as so-called
ket-functions. Whenever we have such a set of vectors, we can set up a complementary set of so-called bra-functions, denoted as f k |. The scalar product of a bra and a
ket yields a number. It is denoted as the bracket: f k |f l . In other words, when a bra
collides with a ket on its right, it yields a scalar number. A bra-vector is completely
defined when its scalar product with every ket-vector of the vector space is given.
A.J. Ceulemans, Group Theory Applied to Chemistry, Theoretical Chemistry and
Computational Modelling, DOI 10.1007/978-94-007-6863-5_2,
© Springer Science+Business Media Dordrecht 2013
11
Function Spaces and Matrices
Abstract This chapter refreshes such necessary algebraic knowledge as will be
needed in this book. It introduces function spaces, the meaning of a linear operator, and the properties of unitary matrices. The homomorphism between operations
and matrix multiplications is established, and the Dirac notation for function spaces
is defined. For those who might wonder why the linearity of operators need be considered, the final section introduces time reversal, which is anti-linear.
Contents
2.1
Function Spaces ...................................
1 1
2 . 2
L i n e a rO p e r a t o r sa n dT r a n s f o r m a t i o nM a t r i c e s ...................
1 2
2 . 3
U n i t a r yM a t r i c e s ...................................
1 4
2 . 4
T i m eR e v e r s a la sa nA n t i - l i n e a rO p e r a t o r ......................
1 6
2 . 5
P r o b l e m s .......................................
1 9
R e f e r e n c e s...........................................
1 9
2.1 Function Spaces
In the first chapter, we saw that if we wanted to rotate the 2p x function, we automatically also needed its companion 2p y function. If this is extended to out-of-plane
rotations, the 2p z function will also be needed. The set of the three p-orbitals forms
a prime example of what is called a linear vector space. In general, this is a space
that consists of components that can be combined linearly using real or complex
numbers as coefficients. An n-dimensional linear vector space consists of a set of n
vectors that are linearly independent. The components or basis vectors will be denoted as f l , with l ranging from 1 to n. At this point we shall introduce the Dirac notation [1] and rewrite these functions as |f l , which characterizes them as so-called
ket-functions. Whenever we have such a set of vectors, we can set up a complementary set of so-called bra-functions, denoted as f k |. The scalar product of a bra and a
ket yields a number. It is denoted as the bracket: f k |f l . In other words, when a bra
collides with a ket on its right, it yields a scalar number. A bra-vector is completely
defined when its scalar product with every ket-vector of the vector space is given.
A.J. Ceulemans, Group Theory Applied to Chemistry, Theoretical Chemistry and
Computational Modelling, DOI 10.1007/978-94-007-6863-5_2,
© Springer Science+Business Media Dordrecht 2013
11